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The Book of Numbers

John Horton Conway and Richard K. Guy
Copernicus Books (Springer)
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on

Just as there are books of special functions, this is a book of special numbers. That’s an oversimplification, and it’s hard to describe the boundaries of this deliberately discursive book, but that does give you the flavor.

Most of the book deals with particular number sequences such as the primes or the Stirling numbers, with particular numbers such as square root of 2 or π, and particular shapes of numbers such as figurate numbers and algebraic numbers, but it also deals with number structures such as complex numbers, infinitesimals, and surreal numbers.

The book starts with a charming chapter on number words: English words that have numbers buried in them (my favorites: biscuit = twice cooked, and uncial = inch-high letters, from Latin uncia = a one-twelfth part). It then takes a meandering tour through all kinds of numbers, darting off into other topics when they naturally come up. For example, the Fibonacci numbers lead to phyllotaxis, and the algebraic numbers lead to the classical Greek ruler-and-compass construction problems.

The announced aim of the book is “to bring to the inquisitive reader without particular mathematical background an explanation of the multitudinous ways in which the word ‘number’ is used.” I wasn’t convinced that this goal would be reached. The mathematical formalism of the book is very small: really nothing beyond high school algebra and trigonometry. But the level of mathematical reasoning is often quite high, particularly in the combinatorial and geometric topics, and I doubt that the average reader, no matter how inquisitive, would be able to work through all of this unaided.

I think the happiest readers for this book would be mathematicians who know some, but not all, of the material presented here. They will be continually surprised and pleased by the connections the book shows.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.


1. The Romance of Numbers

2. Figures from Figures Doing Arithmetic and Algebra by Geometry

3. What Comes Next?

4. Famous Families of Numbers

5. The Primacy of Primes

6. Further Fruitfulness of Fractions

7. Geometric Problems and Algebraic Numbers

8. Imagining Imaginary Numbers

9. Some Transcendental Numbers

10. Infinite and Infinitesimal Numbers